This is a useful model that explains how groups of cars move and how stoplights affect their movement. It was developed by engineers at the Los Alamos Labs and General Motors.  The equations serve to conserve the momentum of the cars and show how the density of the traffic effects the speed at which the cars travel.  To begin, imagine that this is a road with a variable number of cars, from none up to the maximum carrying capacity of the road.
 
 

    The amount of cars on the stretch from x to x+h can be modeled with an equation p(x, t) which gives the density of the cars, if x is the position of the cars and t is the time.

    Taking the integral of this area gives us the number of cars in the area.  The function q is the flux, or the munber of cars going in and out of the area.  The flux of x minus the flux of x+h is used because we want the number of cars going past x minus the number of cars going out of x+h.
 
 

Taking the limit as h approaches 0 gives the equations:

The chain rule can be applied to give:

Which rearranges to:

To go any further with these equations, two assumptions must be made:

        1. Measuring the velocity and position of each individual car on the road is far to difficult.  So, this model views the distribution of cars by looking at the density of the cars, which is the number of cars on the road per mile of road.  We assume that the density is the only property of the cars which matters.
        2. Thesecond assumption follows from the first.  Only the density of the cars matters.  Therefore the average velocity of the cars at any point depends on the density of the cars.  So if v is the function for the velocity of the cars, v = v(p).The flux, q, is also dependent on the density, p, because q = p*v(p). This means that the flux is equal to the density times the velocity, and the velocity is also dependent on the density. So q = q(p).

    The next logical deduction made to solve this model is the realization that these equations are very similiar to the equations used for fluid dynamics.  The cars are almost like a fluid flowing through the enclosed space of the road.  So, the same techniques for solving fluid dynamics problems can be applied to this problem.  Using the Method of Characteristics, we look for lines along which the density is constant.  Taking the derivative of the density function p(x,t) gives:

Where x(t) is the characteristic.  Comparing equation 5 with equation 4 shows that along the characteristic:

Integrating equation 7 in light of equation 8 gives:

This has reduced the problem to a much simpler function.  We can draw a graph of this:

This means that given a density at a time, we can trace along the characteristic lines where p is constant and find the initial density at x₀.
 

The applications of these equations can be seen in the following diagrams:

If the density of all the cars starts out the same, then the traffic flow would stay constant like this:

However, all the cars do not go at the exact same speed. The differences in the velocities of the cars creates a change in the density of the cars.  The density increases in the middle as the cars in the front speed through faster and the cars in the middle become more densely packed together.

Soon, a compression forms:

This because the cars in the back speed up until they reach the more densely  packed region in the middle, in which case they slow down to avoid hitting the other cars.  These cars then become part of the middle pack of cars.  The cars in the front of the pack speed away as the density of the cars around them decreases and they move out of the middle pack.  Thus, the density in the front decreases sharply. Eventually, a gap can open up in the densities, and a shock forms.

Another graphical representation of a shock is when the lines in the graph of the characteristic polynomial cross.  In a system where the density starts out evenly everywhere, then p is a always equal to some constant c, and all the lines along which p is consttn will have the same slope, c, and be parallel.  Thus a shock will never form.  But, a shock can form if p * c for every line.  Then:

This is a discontinuity, and means that the cars coming into the area hit the compression site and have to slow down quickly.

    This model works well, and its accuracy can be seen in every drive through 5 o'clock traffic.  This model helps traffic engineers make the roads easier to drive on, and it also helps them to judge where to put stop lights so as to stop the least number of cars.

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